Kourovka Notebook Problem 9.35

Definitions and mathematical statement

For a topological group $G$, call $G$ inductively compact if \[ \forall F\subseteq G\text{ finite},\quad \exists K\leq G\quad \left(F\subseteq K\ \text{and}\ K\text{ is compact in the subspace topology}\right). \] The source asks whether this property is preserved under lattice isomorphisms between locally compact groups. The subgroup lattice intended by the source is not specified. Two possible readings are $\operatorname{Sub}(G)$, the lattice of all subgroups ordered by inclusion, and $\operatorname{ClSub}(G)$, the lattice of closed subgroups ordered by inclusion.

For a fixed choice $\Lambda=\operatorname{Sub}$ or $\Lambda=\operatorname{ClSub}$, the corresponding precise question is: for every pair of locally compact topological groups $G,H$, if there is a bijection \[ \Phi:\Lambda(G)\longrightarrow\Lambda(H),\qquad A\subseteq B\ \Longleftrightarrow\ \Phi(A)\subseteq\Phi(B) \quad(A,B\in\Lambda(G)), \] does inductive compactness of $G$ imply inductive compactness of $H$? The choice of lattice remains unresolved; these are alternative readings, not two separately attributed conjectures. The supplied record also does not specify whether the term locally compact group includes a Hausdorff assumption, so that convention requires confirmation before fixing the domain.

Short English statement

If every finite subset of a locally compact group lies in a compact subgroup, must the same hold for every locally compact group with an isomorphic subgroup lattice? The source leaves the intended subgroup lattice and the Hausdorff convention unspecified.

Sources

• Local UnsolvedMath registry: permanent ID $700568$, exact source code catalogue entry, title Kourovka Notebook Problem 9.35; supplied upstream revision d371ee917ac9647fbb06cd6fc529deec6f3dda82.
• Attributed to Yu. N. Mukhin, proposed in 1984. The supplied background identifies The Kourovka Notebook (Unsolved Problems in Group Theory), 9th issue (1984), Problem 9.35, and the 21st issue edition (2026), arXiv:1401.0300v48.
• The supplied collection description names E. I. Khukhro and V. D. Mazurov as editors. The registry labels this problem open; this is a supplied status, not an independently verified literature assessment. No source was independently consulted for this draft.